This article focuses on the compressed representations of the pictures

Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The high-rate asymptotic performance of the quantizer is also studied. We evaluate the rate-distortion performance of the quantizer and compare it to known information-theoretic bounds. The high-rate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, high-rate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.

The problem of designing a multiple description vector quantizer with lattice codebook # is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A 2 and Z i , i = 1, 2, 4, 8 that make use of this labeling algorithm. The high-rate squared-error distortions for this family of vector quantizers are then analyzed for a memoryless source with probability density function p and di#erential entropy h(p) < #. For any a # (0, 1) and rate pair (R, R), it is shown that the two-channel distortion d 0 and the channel 1 (or channel 2) distortion d s satisfy lim R## d 0 2 2R(1+a) = 1 4 G(#)2 2h(p) and lim R## d s 2 2R(1-a) = G(S L )2 2h(p) , where G(#) is the normalized second moment of a Voronoi cell of the lattice # and G(S L ) is the normalized second moment of a sphere in L dimensions. Index Terms: Source Coding, Quantization...

In this paper we present an approach to the generalized Multiple Description problem that is fundamentally different from previously published algorithms. Our approach uses explicit channel coding in the form of Unequal Loss Protection to obtain a solution that incorporates many important properties: it can be used with any progressive source coder; it generates a balanced encoding with information equally dispersed among the descriptions; it adds a quantifiable amount of redundancy; it adapts that amount of redundancy to expected channel conditions; and it can optimize for different distortion measures. These properties allow the system to gradually improve image quality as the number of received descriptions increases. We compare our system to previously published results and show that forward error correction in Multiple Description coding can surpass them by a significant margin. 1. Introduction In generalized Multiple Description (MD) coding [1], N descriptions of a source are...

A multiple description lattice vector quantization technique for two descriptions was recently introduced that uses fine and coarse codebooks that are both lattices. The encoding begins with quantization to the nearest point in the fine lattice. This encoding is an inherent optimization for the decoder that receives both descriptions; performance can be improved with little increase in complexity by considering all decoders in the initial encoding step. Since the altered encoding uses only the lattice structure of the coarse codebook, the fine codebook can be optimized without a lattice constraint, improving performance further. Index terms multiple description source coding, lattice vector quantization, codebook optimization Corresponding author: Vivek Goyal e-mail: v.goyal@ieee.org Bell Labs, Room 2C-178 voice: +1 908 582 6484 600 Mountain Avenue fax: +1 908 582 3340 Murray Hill, NJ 07974-0636 Contents I Introduction 2 I-A Multiple Description Coding . . . . . . . . . ....

Multiple description lattice vector quantization (MDLVQ) is a technique for two-channel multiple description coding. We observe that MDLVQ, in the form introduced by Servetto, Vaishampayan and Sloane in 1999, is inherently optimized for the central decoder; i.e., for a zero probability of a lost description. With a nonzero probability of description loss, performance is improved by modifying the encoding rule (using nearest neighbors with respect to “multiple description distance”) and by perturbing the lattice codebook. The perturbation maintains many symmetries and hence does not significantly effect encoding or decoding complexity. An extension to more than two descriptions with attractive decoding properties is outlined. 1

We derive analytical expressions for the central and side quantizers in an n-channel symmetric multiple-description lattice vector quantizer which, under high-resolution assumptions, minimize the expected distortion subject to entropy constraints on the side descriptions for given packet-loss probabilities. The performance of the central quantizer is lattice dependent whereas the performance of the side quantizers is lattice independent. In fact the normalized second moments of the side quantizers are given by that of an L-dimensional sphere. Furthermore, our analytical results reveal a simple way to determine the optimum number of descriptions. We verify theoretical results with numerical experiments and show that with a packet-loss probability of 5%, a gain of 9.1 dB in MSE over state-of-the-art two-description systems can be achieved when quantizing a two-dimensional unit-variance Gaussian source using a total bit budget of 15 bits/dimension and using three descriptions. With 20 % packet loss, a similar experiment reveals an MSE reduction of 10.6 dB when using four descriptions. 1

The optimization criterion and a practically feasible new algorithm is stated for the optimization of the index assignments of a multiple description unconstrained vector quantizer with an arbitrary number of descriptions. In the simulations, the index-optimized multiple description vector quantizer achieves significant gains in source SNR over scalar multiple description schemes.

Abstract — In this paper we derive analytical expressions for the central and side quantizers which, under high-resolutions assumptions, minimize the expected distortion of a symmetric multiple-description lattice vector quantization (MD-LVQ) system subject to entropy constraints on the side descriptions for given packet-loss probabilities. We consider a special case of the general n-channel symmetric multiple-description problem where only a single parameter controls the redundancy tradeoffs between the central and the side distortions. Previous work on two-channel MD-LVQ showed that the distortions of the side quantizers can be expressed through the normalized second moment of a sphere. We show here that this is also the case for three-channel MD-LVQ. Furthermore, we conjecture that this is true for the general nchannel MD-LVQ. For given source, target rate and packet-loss probabilities we find the optimal number of descriptions and construct the MD-LVQ system that minimizes the expected distortion. We verify theoretical expressions by numerical simulations and show in a practical setup that significant performance improvements can be achieved over state-of-the-art two-channel MD-LVQ by using three-channel MD-LVQ. Index Terms — high-rate quantization, lattice quantization, multiple description coding, vector quantization. I.

iii To my parents, my brother, and my dear wife Acknowledgements iv First of all, I would like to express my heartfelt gratitude to my advisor, Prof. Richard M. Murray, for his guidance that is so conducive to the work I have undertaken. His broad knowledge, deep insights, outstanding leadership, and great personality make him a mentor that every student dreams of having. I am grateful to my committee members: Prof. John C. Doyle, Prof. Joel W. Burdick, Prof. Babak Hassibi, and Prof. Tracey Ho, for inspiring discussions and detailed comments on my thesis. I also appreciate the help from Prof. Michelle Effros, Prof. Leonard Schulman and Prof. Steven Low. There are many other people I would like to acknowledge with thankfulness for the help given to me in the last five years. Thanks to Dr. Alcherio Martinoli for the unforgettable support during my first year at Caltech. Thanks to Dr. Reza Olfati-Saber for showing me his wonderful work on coordinated control. Thanks to Dr. Eric Klavins, Dr. Lars Cremean, David van Gogh, and Steve Waydo for helping me conduct experiments on the testbed. Also, thanks to Vijay Gupta, Ling Shi, Abhishek Tiwari, and other group members for collaborations and brainstorming. Because of them, my research became much easier and more fruitful. Also thanks to staff members at Caltech for making my administrative life easy.